This book studies number theory the old-fashioned way — by looking at lots of numbers. Most of the sections start out with a big table of numbers. For example, to start the discussion of the Diophantine equation x2 + y2 = z2, we have a whole page listing the squares and using that to form the numbers that are sums of two squares. We are then invited to circle all the squares in the table and look for a pattern.

The style of the book is similar to Pólya and Szegö’s Problems and Theorems in Analysis. The book consists of many long sequences of problems, each leading to a notable result, and there are very brief solutions in the back of each chapter. The Burn book is much easier that Polya & Szego, partly because it does not go very deep and partly because there are many more steps. The pace seems very slow to someone who is already familiar with the material, but I think it does give a good feel for how research is really done, with lots of experimentation and wandering around.

The book makes heavier use of drawings and graphs than is typical in beginning number theory books. The topics are typical of beginning number theory books, with a few surprises such as quadratic forms and quite a lot on the geometry of numbers.

Pathway is advertised as requiring only a high-school background. This is a slight exaggeration as it makes some use of abstract algebra. It also makes some use of complex numbers and quaternions.

The book’s first edition was in 1982, but the book is what we would today call “inquiry-based.” I compared it to the 2007 volume Number Theory Through Inquiry by Marshall, Odell, and Starbird. The latter book is much more conventional; in fact to some extent it looks like a plain number theory book with definitions, theorems, and examples, but with the proofs left out. The latter authors are also very interested in pedagogy and they have a lot of tips and exercises to see if you are learning the material; Burn is only interested in numbers.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.